# Learning with Holographic Reduced Representations

arXiv: https://arxiv.org/abs/2109.02157 · HTML: https://mlsanity.com/paper/2109.02157/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 60 references that are in the map.
- First posted on arXiv 5 Sep 2021.

## Builds on (most cited first, 22 of 22)

- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [The Neuro-Symbolic Concept Learner: Interpreting Scenes, Words, and Sentences From Natural Supervision](https://mlsanity.com/paper/1904.12584/) (2019)
- [A Metric Learning Reality Check](https://mlsanity.com/paper/2003.08505/) (2020)
- [On the Binding Problem in Artificial Neural Networks](https://mlsanity.com/paper/2012.05208/) (2020)
- [Accounting for Variance in Machine Learning Benchmarks](https://mlsanity.com/paper/2103.03098/) (2021)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Are We Really Making Much Progress? A Worrying Analysis of Recent Neural Recommendation Approaches](https://mlsanity.com/paper/1907.06902/) (2019)
- [Holographic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/1510.04935/) (2015)
- [AttentionXML: Label Tree-based Attention-Aware Deep Model for High-Performance Extreme Multi-Label Text Classification](https://mlsanity.com/paper/1811.01727/) (2018)
- [DiSMEC - Distributed Sparse Machines for Extreme Multi-label Classification](https://mlsanity.com/paper/1609.02521/) (2016)
- [A comparison of Vector Symbolic Architectures](https://mlsanity.com/paper/2001.11797/) (2020)
- [LightXML: Transformer with Dynamic Negative Sampling for High-Performance Extreme Multi-label Text Classification](https://mlsanity.com/paper/2101.03305/) (2021)
- [Large-scale Multi-label Learning with Missing Labels](https://mlsanity.com/paper/1307.5101/) (2013)
- [A no-regret generalization of hierarchical softmax to extreme multi-label classification](https://mlsanity.com/paper/1810.11671/) (2018)
- [Associative Long Short-Term Memory](https://mlsanity.com/paper/1602.03032/) (2016)
- [Extreme Classification in Log Memory using Count-Min Sketch: A Case Study of Amazon Search with 50M Products](https://mlsanity.com/paper/1910.13830/) (2019)
- [DECAF: Deep Extreme Classification with Label Features](https://mlsanity.com/paper/2108.00368/) (2021)
- [A theory of sequence indexing and working memory in recurrent neural networks](https://mlsanity.com/paper/1803.00412/) (2018)
- [Learning to Reason with Third-Order Tensor Products](https://mlsanity.com/paper/1811.12143/) (2018)
- [CircConv: A Structured Convolution with Low Complexity](https://mlsanity.com/paper/1902.11268/) (2019)
- [Accelerating Extreme Classification via Adaptive Feature Agglomeration](https://mlsanity.com/paper/1905.11769/) (2019)
- [SSH (Sketch, Shingle, & Hash) for Indexing Massive-Scale Time Series](https://mlsanity.com/paper/1610.07328/) (2016)

## Built on by (most cited first, 25 of 25)

- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges](https://mlsanity.com/paper/2112.15424/) (2021)
- [Torchhd: An Open Source Python Library to Support Research on Hyperdimensional Computing and Vector Symbolic Architectures](https://mlsanity.com/paper/2205.09208/) (2022)
- [Recasting Self-Attention with Holographic Reduced Representations](https://mlsanity.com/paper/2305.19534/) (2023)
- [A Brain-Inspired Low-Dimensional Computing Classifier for Inference on Tiny Devices](https://mlsanity.com/paper/2203.04894/) (2022)
- [What Do Machine Learning Researchers Mean by "Reproducible"?](https://mlsanity.com/paper/2412.03854/) (2024)
- [Deploying Convolutional Networks on Untrusted Platforms Using 2D Holographic Reduced Representations](https://mlsanity.com/paper/2206.05893/) (2022)
- [Factorizers for Distributed Sparse Block Codes](https://mlsanity.com/paper/2303.13957/) (2023)
- [Towards Generalization in Subitizing with Neuro-Symbolic Loss using Holographic Reduced Representations](https://mlsanity.com/paper/2312.15310/) (2023)
- [Vector-Symbolic Architecture for Event-Based Optical Flow](https://mlsanity.com/paper/2405.08300/) (2024)
- [A Walsh Hadamard Derived Linear Vector Symbolic Architecture](https://mlsanity.com/paper/2410.22669/) (2024)
- [Lempel-Ziv Networks](https://mlsanity.com/paper/2211.13250/) (2022)
- [Recursive Binding for Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2201.11691/) (2022)
- [Optimal quadratic binding for relational reasoning in vector symbolic neural architectures](https://mlsanity.com/paper/2204.07186/) (2022)
- [Linearithmic Clean-up for Vector-Symbolic Key-Value Memory with Kroneker Rotation Products](https://mlsanity.com/paper/2506.15793/) (2025)
- [Multi-label Learning with Random Circular Vectors](https://mlsanity.com/paper/2407.05656/) (2024)
- [FoGE: Fock Space inspired encoding for graph prompting](https://mlsanity.com/paper/2507.02937/) (2025)
- [Efficient Vector Symbolic Architectures from Histogram Recovery](https://mlsanity.com/paper/2511.01838/) (2025)
- [A Vector Symbolic Approach to Multiple Instance Learning](https://mlsanity.com/paper/2511.16795/) (2025)
- [Reinforcement Learning Enhancement Using Vector Semantic Representation and Symbolic Reasoning for Human-Centered Autonomous Emergency Braking](https://mlsanity.com/paper/2602.05079/) (2026)
- [Disentanglement with Holographic Reduced Representations](https://mlsanity.com/paper/2606.09725/) (2026)
- [Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories](https://mlsanity.com/paper/2606.11391/) (2026)
- [Gram-Space: Structure-Preserving Codebook Compression for Memory-Efficient Neuro-Symbolic AI](https://mlsanity.com/paper/2608.01528/) (2026)
- [Learning a Vector-Symbolic Model for Socio-Cultural Tasks](https://mlsanity.com/paper/2608.02807/) (2026)
- [Learning Holographic Reduced Representations with Clifford Variational Autoencoders](https://mlsanity.com/paper/2609.28409/) (2026)

## Abstract (arXiv)

Holographic Reduced Representations (HRR) are a method for performing symbolic AI on top of real-valued vectors by associating each vector with an abstract concept, and providing mathematical operations to manipulate vectors as if they were classic symbolic objects. This method has seen little use outside of older symbolic AI work and cognitive science. Our goal is to revisit this approach to understand if it is viable for enabling a hybrid neural-symbolic approach to learning as a differentiable component of a deep learning architecture. HRRs today are not effective in a differentiable solution due to numerical instability, a problem we solve by introducing a projection step that forces the vectors to exist in a well behaved point in space. In doing so we improve the concept retrieval efficacy of HRRs by over 100×. Using multi-label classification we demonstrate how to leverage the symbolic HRR properties to develop an output layer and loss function that is able to learn effectively, and allows us to investigate some of the pros and cons of an HRR neuro-symbolic learning approach. Our code can be found at https://github.com/NeuromorphicComputationResearchProgram/Learning-with-Holographic-Reduced-Representations